LMFDB - Newform orbit 42.12.a.b

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [42,12,Mod(1,42)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("42.1"); S:= CuspForms(chi, 12); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(42, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 12, names="a")

Level:	$ N $	$=$	$ 42 = 2 \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	42.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,-32,-243,1024,-4580] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$32.2704135835$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$

$=$

$ q - 32 q^{2} - 243 q^{3} + 1024 q^{4} - 4580 q^{5} + 7776 q^{6} - 16807 q^{7} - 32768 q^{8} + 59049 q^{9} + 146560 q^{10} - 522754 q^{11} - 248832 q^{12} - 225726 q^{13} + 537824 q^{14} + 1112940 q^{15}+ \cdots - 30868100946 q^{99}+O(q^{100}) $

q - 32 * q^2 - 243 * q^3 + 1024 * q^4 - 4580 * q^5 + 7776 * q^6 - 16807 * q^7 - 32768 * q^8 + 59049 * q^9 + 146560 * q^10 - 522754 * q^11 - 248832 * q^12 - 225726 * q^13 + 537824 * q^14 + 1112940 * q^15 + 1048576 * q^16 - 7106092 * q^17 - 1889568 * q^18 - 10905820 * q^19 - 4689920 * q^20 + 4084101 * q^21 + 16728128 * q^22 - 9537514 * q^23 + 7962624 * q^24 - 27851725 * q^25 + 7223232 * q^26 - 14348907 * q^27 - 17210368 * q^28 - 4122410 * q^29 - 35614080 * q^30 + 225621072 * q^31 - 33554432 * q^32 + 127029222 * q^33 + 227394944 * q^34 + 76976060 * q^35 + 60466176 * q^36 + 401516714 * q^37 + 348986240 * q^38 + 54851418 * q^39 + 150077440 * q^40 - 449235192 * q^41 - 130691232 * q^42 + 862042724 * q^43 - 535300096 * q^44 - 270444420 * q^45 + 305200448 * q^46 + 2301032388 * q^47 - 254803968 * q^48 + 282475249 * q^49 + 891255200 * q^50 + 1726780356 * q^51 - 231143424 * q^52 + 4033367010 * q^53 + 459165024 * q^54 + 2394213320 * q^55 + 550731776 * q^56 + 2650114260 * q^57 + 131917120 * q^58 - 4862200904 * q^59 + 1139650560 * q^60 + 5473241454 * q^61 - 7219874304 * q^62 - 992436543 * q^63 + 1073741824 * q^64 + 1033825080 * q^65 - 4064935104 * q^66 - 5274256280 * q^67 - 7276638208 * q^68 + 2317615902 * q^69 - 2463233920 * q^70 - 13847661862 * q^71 - 1934917632 * q^72 - 12935724098 * q^73 - 12848534848 * q^74 + 6767969175 * q^75 - 11167559680 * q^76 + 8785926478 * q^77 - 1755245376 * q^78 + 32414725668 * q^79 - 4802478080 * q^80 + 3486784401 * q^81 + 14375526144 * q^82 - 30154096492 * q^83 + 4182119424 * q^84 + 32545901360 * q^85 - 27585367168 * q^86 + 1001745630 * q^87 + 17129603072 * q^88 - 57844151544 * q^89 + 8654221440 * q^90 + 3793776882 * q^91 - 9766414336 * q^92 - 54825920496 * q^93 - 73633036416 * q^94 + 49948655600 * q^95 + 8153726976 * q^96 + 70497350734 * q^97 - 9039207968 * q^98 - 30868100946 * q^99

Embeddings

For each embedding $\iota_m$ of the coefficient field, the values $\iota_m(a_n)$ are shown below.

For more information on an embedded modular form you can click on its label.

comment:embeddings in the coefficient field

gp:mfembed(f)

Label

$\iota_m(\nu)$

$ a_{2} $

$ a_{3} $

$ a_{4} $

$ a_{5} $

$ a_{6} $

$ a_{7} $

$ a_{8} $

$ a_{9} $

$ a_{10} $

1.1

−32.0000

−243.000

1024.00

−4580.00

7776.00

−16807.0

−32768.0

59049.0

146560.

Atkin-Lehner signs

$ p $	Sign
$2$	$ +1 $
$3$	$ +1 $
$7$	$ +1 $

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	42.12.a.b	✓	1
3.b	odd	2	1		126.12.a.g		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
42.12.a.b	✓	1	1.a	even	1	1	trivial
126.12.a.g		1	3.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5} + 4580 $ T5 + 4580 acting on $S_{12}^{\mathrm{new}}(\Gamma_0(42))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T + 32 $ T + 32
$3$	$ T + 243 $ T + 243
$5$	$ T + 4580 $ T + 4580
$7$	$ T + 16807 $ T + 16807
$11$	$ T + 522754 $ T + 522754
$13$	$ T + 225726 $ T + 225726
$17$	$ T + 7106092 $ T + 7106092
$19$	$ T + 10905820 $ T + 10905820
$23$	$ T + 9537514 $ T + 9537514
$29$	$ T + 4122410 $ T + 4122410
$31$	$ T - 225621072 $ T - 225621072
$37$	$ T - 401516714 $ T - 401516714
$41$	$ T + 449235192 $ T + 449235192
$43$	$ T - 862042724 $ T - 862042724
$47$	$ T - 2301032388 $ T - 2301032388
$53$	$ T - 4033367010 $ T - 4033367010
$59$	$ T + 4862200904 $ T + 4862200904
$61$	$ T - 5473241454 $ T - 5473241454
$67$	$ T + 5274256280 $ T + 5274256280
$71$	$ T + 13847661862 $ T + 13847661862
$73$	$ T + 12935724098 $ T + 12935724098
$79$	$ T - 32414725668 $ T - 32414725668
$83$	$ T + 30154096492 $ T + 30154096492
$89$	$ T + 57844151544 $ T + 57844151544
$97$	$ T - 70497350734 $ T - 70497350734
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Classical	Maass
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Elliptic curves over $\Q$
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Genus 2 curves over $\Q$
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Belyi maps

Level:	\( N \)	\(=\)	\( 42 = 2 \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	42.a (trivial)

\(n\):		e.g. 2-40 or 990-1000
Significant digits:
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